B-spline differentiation formula ($N_{i,d}^{\prime }=\frac{d}{t_{i+d}-t_i}{N}_{i,d-1}-\frac{d}{t_{i+d+1}-t_{i+1}}{N}_{i+1,d-1}$)

For the partition-of-unity normalization, the derivative of a degree-$d$ B-spline is the displayed difference of two overlapping degree-$(d-1)$ B-splines. A zero denominator denotes a term with collapsed support, assigned zero. For simple knots the divided-difference representation $N_{i,d}(t)=(t_{i+d+1}-t_i)[t_i,\dots ,t_{i+d+1}](\tau -t{)}_{+}^d$ has the required support, smoothness and integral normalization. Differentiate the truncated power and apply the last-step divided difference recursion to obtain the formula. Repeated-knot versions follow by confluent limits, with one-sided interpretation where necessary.